Optimal Control of Information Epidemics
Aditya Karnik and Pankaj Dayama
Global General Motors R&D
India Science Lab, GM Tech Center (India)
Bangalore 560066, India
Email: {aditya.karnik,pankaj.dayama}@gm.com
Abstract—In this work we address two problems concerning
information propagation in a population: a) how to maximize the
spread of a given message in the population within the stipulated
time and b) how to create a given level of buzz- measured by
the fraction of the population engaged in conversation on a topic
of interest- at a specified time horizon. Arising in the context of
two rather disparate networks- social and wireless vehicular networks, their importance in campaigning on social networks and
security in vehicular networks can only be understated. Taking
a mean-field route, we pose the two problems as continuous-time
deterministic optimal control problems. We characterize optimal
controls, present some numerical results and provide practical
insights. Interestingly, the problems we address are antithesis to
those in disease epidemics, which need to be contained rather
than actively spread.
I. I NTRODUCTION
The advent of Web 2.0 technologies and an explosion in
social media applications in recent years have revolutionized
social networks. While these networks have existed since time
immemorial, technological advances have made it possible for
people to connect, communicate and share with each other
easily, (almost) instantaneously, using a variety of media,
from anywhere, anytime and without geographical or cultural
boundaries. Social networks have also expanded in scope to
include brands, businesses and news media in peoples’ on-line
social circles. It is, therefore, not surprising that campaignersbe them social, political or marketing- find (on-line) social
networks attractive forums to reach out to masses1 . Campaigners, typically, have two main concerns among others2 : a) to be
able to communicate a piece of information, say, a marketing
message, to as large a fraction of the population as possible,
in stipulated time, and b) to be able to engage people in
conversations on a topic (which entails exchange of multiple
messages on the same subject but not necessarily the same
content) so that by a given time the impact of promotional
activities (“buzz”) reaches a desired level. Neither task is easy
however, since with the huge amount of information generated
today, it is difficult to get eyeballs on specific messages and
retain peoples’ interest.
Similar concerns arise in vehicular networks (or more generally, wireless sensor networks) as well. Wireless vehicle-tovehicle (V2V) communications aim to improve driver safety,
1 See numerous blogs devoted to discussing these issues, most notably
Mashable, Socialsignal and Internet marketing blog.
2 To be able to contain rumors/negative news is one such.
c 2012 IEEE
978-1-4673-0298-2/12/$31.00 by having vehicles exchange messages pertaining to kinematic
parameters, road condition and traffic information [1]. Alerts
arising out of, say, a hazardous road condition need to be
conveyed to vehicles approaching the target area by certain
time to be of any use to the drivers. Vehicular networks also
need to support dissemination of system-wide alerts/updates
from a security infrastructure (such as based on public key
cryptography (PKI)), which ensures that the messages acted
upon by the safety critical V2V applications are protected
from spoofing, alteration, and replay. One-time updates (e.g.,
fixes, queries) are required to reach a large fraction of vehicles
in a given time whereas recurrent updates (e.g., certificate
revocation lists [2]) need to be disseminated in such way that
system’s vulnerability to attacks is as low as possible. Here
again the two tasks are not easy given the expected volume
of messages/alerts, limited infrastructure and bandwidth for
transmissions and idiosyncrasies of the wireless channel.
In this paper, we address the two problems described above
in a rather general setting. We consider a population of “smart”
agents (i.e., people, vehicles or sensors) with an underlying
network specifying conduits for information (or message)
transfer. Each agent is a potential producer, consumer and relay
of messages. Message propagation is epidemic in nature, i.e.,
if an agent communicates a message to its neighbors, they may
in turn communicate it to their neighbors and so on. Agents
can also be communicated messages directly and recruited for
spreading them, albeit at a cost. In the case of a social networks
this cost may be seen as the cost of incentives offered to people
to forward messages to their friends whereas in a vehicular
network it could be the cost of using fixed infrastructure
(e.g., road side units) to communicate with the vehicles. A
budgetary constraint implies that not all agents can be given
message(s) directly. This necessitates an optimal control of
recruitment and, thereby, epidemic transmission processes in
the population. We pose the two problems described earlier as
follows.
P1 Message dissemination: Given a budget constraint, maximize the fraction of the population receiving a tagged
message by the stipulated time, by controlling the number
of recruits over time.
P2 Topic diffusion: Minimize the cost incurred to create
a given level of buzz, i.e., the fraction of population
engaged in conversion on a topic, at a specified time,
by controlling the number of recruits over time.
To make the problems mathematically precise, we take
the mean-field approach and obtain a system of ordinary
differential equations (ODEs) to describe the system evolution.
Given that the networks under consideration are large, we
believe that the thermodynamic limit (i.e., taking the size of
the population to infinity) to obtain the system of ODEs is
justified and the controls we derive have practical significance.
In order to simplify our analyses and to deliver key insights,
in this paper, we will underplay the structure of the networks
and characterize them only through the average degree3 . We
are, therefore, led to the SIR epidemic system for problem
P1 and the SIS epidemic system for problem P2. The two
problems are then formulated as continuous-time deterministic
optimal control problems. We characterize optimal controls
using Pontryagin Maximum principle, validate them through
numerical studies and provide some practical insights.
We show that in the case of problem P1 (message dissemination), the optimal control has switching behavior- it is optimal
to recruit for some initial duration till the budget is spent
and then stop. In problem P2 (topic diffusion), the optimal
strategy shows different behaviors, depending on whether the
population can sustain interest in the topic (this is formalized
in terms of the stability of a certain equilibrium). If the topic
cannot be sustained, then it is optimal to start recruiting just
before the desired time horizon. On the other hand, if the
topic can be sustained, then the optimal strategy is to recruit
for some initial duration and then stop.
This paper is organized as follows. Related work is reviewed
in Section II. In Section III and Section IV we discuss the message dissemination problem and the topic diffusion problem
respectively. Numerical results are provided in Section V. We
conclude in Section VI.
II. R ELATED W ORK
Our work is closely related to epidemic processes for which
there is a long history of work [4]. Not surprisingly, there is
also a significant body of work on the control of epidemics
given its practical importance; see, for example, [5], [6] for
some optimal control formulations. Epidemic models and
control formulations have also been investigated in the context
of on-line viruses and worms (see [7] and reference therein).
Though the models we analyze in this paper are essentially
the epidemic models, the problems we address are antithesis
to those in the line work of described above- disease/virus
needs to be contained whereas in our case information is to be
widely spread. Problems concerning maximization of spread of
rumors and malware have attracted attention recently: impulse
control of rumor spreading is studied in [8] and optimal control
of malware propagation assuming a variation of the SIR model
is investigated in [9]. Our work differs from the former both
in the model and the problem formulation (it assumes DaleyKendall and Maki-Thomson models) whereas it differs from
the latter in terms of the control used (it considers ‘kill rate’
3 In
the context of vehicular networks it may be more reasonable to assume
homogeneous mixing directly than imposing a network structure (see [3] and
references therein).
and ‘infection rate’ as controls). Moreover, we also investigate
the control formulation for the SIS system.
In the case of social networks, research has focused more
on understanding information diffusion [10] whereas in the
case of wireless mobile networks the focus is more on design
and evaluation of algorithms/protocols for dissemination (e.g.,
[11])) and distributed computation (see [12] for gossip algorithms). We believe that our problem formulations and control
strategies are first of their kind in the context of social and
vehicular networks.
III. M ESSAGE D ISSEMINATION
Consider a population of N agents. The interconnections
among them are specified by graph G, which is drawn randomly from the set of undirected graphs of size N and
average degree k. The agents become active at random times
to transmit messages. “Activity” process of each agent is
modeled as a Poisson process of rate 1 and is assumed to be
independent of those of other agents. We use epidemiological
terminology and refer to an agent as susceptible if she has
not received the tagged message. An agent becomes infected
when she receives the tagged message and actively spreads
it. An infected agent is said to have recovered if she stops
spreading the tagged message. If the message is communicated
by one of her social contacts, a susceptible agent turns into
an infected one with probability β ′ . In vehicular networks, β ′
is simply the probability of successful reception whereas in
social networks, it can be seen as the probability of conversion
due under social influence [13]. A susceptible agent may also
become infected by direct “recruitment”- when a susceptible
agent becomes active, with probability u, she is sent the tagged
message directly and recruited to act as a spreader at a given
cost. An infected agent recovers with probability v > 0 upon
being active.
Now given a fixed budget B > 0 to recruit agents, the
objective is to exercise control u so that starting with a
susceptible population, the number of recipients of the tagged
message by given time T < ∞ is maximized. In view of
the large scale of the networks under consideration, we take
the following approach. Let S (resp. I) denote the number of
susceptible (resp. infected) agents in the population. Assuming
that agents are indistinguishable except for their states, it follows that (S, I) is a continuous-time Markov chain. Taking a
mean-field limit [14], we obtain the following (SIR epidemic)
system of ODEs, where s (resp. i) denotes the fraction of
susceptible (resp. infected) population, β := β ′ k and u(·) is
the recruitment control4.
ṡ(t)
i̇(t)
=
=
−βs(t)i(t) − u(t)s(t)
βs(t)i(t) − vi(t) + u(t)s(t)
ṙ(t)
=
vi(t)
(1)
(2)
4 While a mean field limit can be established for the controlled Markov
chain on the lines of [15], we do not do so here. We take the mean field limit
of the stochastic system and treat parameter u as control, for controlling the
above system of ODEs.
Observe that ṡ + i̇ + ṙ = 0. Therefore, it suffices to consider
only (1) and (2). Let Ω := {(s, i)|0 ≤ s, i ≤ 1, s + i ≤ 1}
and U := {u|0 ≤ u ≤ ū}. Henceforth we scale ū to 1 without
loss of generality.
The cost of recruitment is linearly proportional to the recruitment effort u. With appropriate
R T normalization the budget
constraint can be expressed as 0 u(t)dt ≤ B. It is assumed
that B < T , otherwise the problem has a trivial solution that
u(t) = 1, ∀t. This isoperimetric constraint can be converted
into a (more amenable) constraint on the terminal state by
defining a (dummy) state variable w(t) with ẇ(t) = −u(t)
and w(0) = B. The budget constraint is then simply stated as
w(T ) ≥ 0.
Let xT (t) := (s(t), i(t), w(t)) denote the state vector and


−βs(t)i(t) − u(t)s(t)


f(x(t), u(t)) :=  βs(t)i(t) − vi(t) + u(t)s(t) 
(3)
−u(t)
The problem can now be formally stated as
Maximize 1 − s(T )
(4)
subject to ẋ(t) = f(x(t), u(t)) and the following constraints
on state and control variables: for all 0 ≤ t ≤ T, x(t) ∈
Ω × R+ , u(t) ∈ U, x(0) = (1, 0, B) and w(T ) ≥ 0.
We say that (x∗ (·), u∗(·)) is an optimal pair of state and
control trajectory if u∗ (·) is piecewise continuous, x∗ (·) is
continuous and piecewise continuously differentiable and u∗ (·)
maximizes the objective (4) under the stated constraints. Our
main result is the following.
Proposition 3.1: u∗ (t) = 1 for 0 ≤ t ≤ B and u∗ (t) = 0
for B < t ≤ T .
In the following we present a proof of Proposition 3.1. We
start by observing from (3) that a unique solution exists in
[0, T ] for any initial condition x(0) ∈ Ω × R+ . Moreover,
Ω is positively invariant: {(s, 0)|0 ≤ s ≤ 1} ∈ Ω is the set
of equilibria; ṡ = 0 whenever s = 0 and on the boundary
s + i = 1, ṡ + i̇ ≤ 0. Therefore, a solution starting from any
initial condition (s, i) ∈ Ω(and in particular (1, 0)) remains
confined to Ω. Hence, the path-wise state constraints can be
ignored from the problem statement.
Note that the formulation involves only terminal cost and
the system (3) is affine in control. Existence of an optimal
control is then established by Filippov-Cesari theorem [16]
which guarantees that the reachable set is compact and by
noting that the cost is continuous function of the state. We
derive the structural form of an optimal control as follows.
Denoting by pT (t) = (p1 (t), p2 (t), p3 (t)) the vector of
adjoint variables, from (3) and (4) the Hamiltonian for our
problem can be written as
Adjoint condition: There exists continuous and piecewise
continuously differentiable function p∗ (·) that solves the following system of ODEs at all t where u∗(·) is continuous.
ṗ1 (t)
=
(p1 (t) − p2 (t))(βi(t) + u(t))
(6)
ṗ2 (t)
ṗ3 (t)
=
=
β(p1 (t) − p2 (t))s(t) + p2 (t)v
0
(7)
(8)
Transversality condition:
p∗1 (T ) = −1, p∗2 (T ) = 0, p∗3 (T ) ≥ 0, p∗3 (T )w ∗ (T ) = 0 (9)
Hamiltonian maximizing condition: For all t ∈ [0, T ] and
u∈U
H(x∗ (t), p∗(t), u∗ (t)) ≥ H(x∗ (t), p∗(t), u)
(10)
From (5) and (10) we get
(
0 (p∗2 (t) − p∗1 (t))s∗ (t) < p∗3 (t)
u∗(t) =
1 (p∗2 (t) − p∗1 (t))s∗ (t) > p∗3 (t)
u∗ (t) takes an arbitrary value in U in case p∗2 (t) −
p∗1 (t))s∗ (t) = p∗3 (t). (8) implies that p3 (t) is constant. Denote it by c and denote by φ(t) the “switching function”
(p∗2 (t) − p∗1 (t))s∗ (t) − c.
Let Ht∗ denote the Hamiltonian at time t along
∗
(x (·), u∗(·)). Then from (9) we get HT∗ = βs∗ (T )i∗ (T ) +
u∗ (T )(s∗ (T ) − c). Now T < ∞, B > 0 and (s(0), i(0)) =
(1, 0). Therefore, s∗ (T ) > 0, i∗ (T ) > 0 and hence HT∗ > 0.
Since the Hamiltonian is constant for autonomous systems,
H0∗ > 0. Hence φ(0) > 0, and so u∗ (0) = 1.
Let τ be such that φ(τ ) = 0. Then from (5) we have
Hτ∗
=
=
β(p∗2 (τ ) − p∗1 (τ ))s∗ (τ )i∗ (τ ) − p∗2 (τ )vi∗ (τ )
cβi∗ (τ ) − p∗2 (τ )vi∗ (τ )
Hτ∗ > 0 implies that p∗2 (τ ) < cβ
v . This, in turn, implies that
ṗ∗2 (τ ) = (−cβ + p∗2 (τ )v) < 0 (see (7)). Note that φ(·) is
continuous and piecewise continuously differentiable. From
(3), (6) and (7) we get
φ̇(τ ) = s∗ (τ )ṗ∗2 (τ ) < 0
This implies that φ(τ ) = 0 for at most one value of τ since
at every “crossing” the slope is negative (there is no singular
interval). Let t∗ denote this value. t∗ < T since B < T . For
T ≥ t > t∗ , φ(t) < 0; hence u∗ (t) = 0. As φ(0) > 0, t∗ > 0
and continuity implies that for 0 ≤ t ≤ t∗ , φ(t) ≥ 0 and hence
u∗ (t) = 1. u∗ (·) is, thus, a bang-bang control. Then clearly
t∗ = B otherwise the budget is underutilized and the solution
is not optimal. The proposition is, thus, established.
IV. TOPIC D IFFUSION
The underlying stochastic model in this case is similar
to
the one described in Section III. The difference is that
H(x(t), p(t), u(t)) = β(p2 (t) − p1 (t))s(t)i(t) − p2 (t)vi(t)
an
infected agent does not recover but goes back to being
+((p2 (t) − p1 (t))s(t) − p3 (t))u(t)(5)
susceptible. In the case of social networks, the interpretation
The Maximum principle [16] provides the following neces- is that during the infected phase, an agent participates in
the conversation (by transmitting messages on the topic). She
sary conditions for (x∗ (·), u∗(·)).
becomes susceptible when she withdraws from the discussion.
She can again be pulled into the discussion, thereby, again
becoming infected. In vehicular networks, being in infected
state may mean that a (pre-designated relay) vehicle has the
latest security update (say, certificate revocation list). Over a
period of time, it becomes outdated and the vehicle is rendered
susceptible to attacks. The reception of the latest update makes
it infected again5 . As in Section III, the agents can be infected
by recruiting them at a cost; when a susceptible agent becomes
active, with probability u, she is recruited to be a participant.
In the following, we use the social network terminology.
The buzz at time t is measured in terms of the fraction of the
population engaged in discussing the topic, i.e., infected. The
objective is to minimize the cost of creating a certain level of
buzz at given time T , the cost being that of recruiting agents.
Taking the same approach as in Section III, we obtain the
following (SIS epidemic) system of ODEs to be controlled
through recruitment effort u(·).
ṡ(t)
=
−βs(t)i(t) + vi(t) − u(t)s(t)
i̇(t)
=
βs(t)i(t) − vi(t) + u(t)s(t)
(11)
Since s(t) + i(t) = 1, ∀t, it suffices to consider only (11).
Replacing s(t) with 1 − i(t), we obtain
i̇(t)
= −βi(t)2 + (β − v − u(t))i(t) + u(t)
(12)
As in Section III the cost of recruitment is taken linearly
proportional to the recruitment effort u and is normalized to
1. Denoting by b > 0 the target level of buzz, the problem is
now stated formally as follows.
Z T
Maximize −
u(t)dt
(13)
0
subject to (12) and the following constraints on state and
control variables: for all 0 ≤ t ≤ T, i(t) ∈ Ω := {i|0 ≤
i ≤ 1}, u(t) ∈ U := {u|0 ≤ u ≤ 1}, i(0) = 0 and i(T ) ≥ b.
Let y1 and y2 denote the roots of i2 + 1+v−β
i − β1 = 0.
β
Proposition 4.1: Buzz level b is achievable at T if and only
if T ≥ β(y21−y1 ) [ln(1 − yb1 ) − ln(1 − yb2 )].
Proposition 4.1 simply states that any buzz level not greater
than that reachable by applying u(t) = 1, ∀t is achievable at
T . We assume that b is achievable at T and that strict inequality
holds in Proposition 4.1 (else the solution is trivial).
Let (i∗ (·), u∗(·)) denote an optimal state and control pair
(optimality as defined in Section III).
Proposition 4.2: 1) If βv > 1, then there exists 0 < t∗ <
T such that u∗ (t) = 0 for 0 ≤ t ≤ t∗ and u∗ (t) = 1 for
t∗ < t ≤ T .
2) If βv < 1, then there exist 0 < t∗ < t̃ ≤ T such that
u∗ (t) = 0qfor t∗ < t ≤ t̃ and u∗(t) = 1 otherwise. If
b ≤ (1 − βv ), t̃ = T . If b ≥ (1 − βv ), t̃ < T .
In the rest of the section, we prove Proposition 4.2. First,
observe that i̇ ≥ 0 for i = 0 and i̇ < 0 for i = 1. Therefore,
Ω is positively invariant and the path-wise state constraint can
5 Thus,
in our formulation, being infected is actually good for health!
be ignored. Second, the terminal state constraint must be met
with equality, i.e., i∗ (T ) = b, for optimality.
The Hamiltonian for this problem is
H(i(t), p(t), p0 , u(t)) =
p(t)(−βi(t)2 + (β − v)i(t))
+(p(t)(1 − i(t)) − p0 )u(t)(14)
p(t) denotes the adjoint variable and p0 is the multiplier. From
the Maximum principle [16] we obtain the following necessary
conditions for (i∗ (·), u∗(·)).
Adjoint condition: There exist p∗0 , p∗(t) ≥ 0, (p∗0 , p∗(t)) 6=
0. p∗(t) is a continuous and piecewise continuously differentiable function that solves the following ODE at all t where
u∗ (·) is continuous.
ṗ∗ (t) = p(t)(2βi(t) + v − β + u(t))
(15)
Transversality condition:
p∗ (T ) ≥ 0, p∗ (T )(i∗ (T ) − b) = 0
Hamiltonian maximizing condition: For all t ∈ [0, T ] and
u∈U
H(i∗ (t), p∗ (t), p∗0 , u∗ (t)) ≥ H(i∗ (t), p∗(t), p∗0 , u)
This yields the following form for the optimal control.
(
0 p∗ (t)(1 − i∗ (t)) < p∗0
∗
u (t) =
1 p∗ (t)(1 − i∗ (t)) > p∗0
u∗ (t) takes an arbitrary value in U in case p∗ (t)(1 − i∗ (t)) =
p∗0 .
If p∗0 = 0 then p∗ (t) > 0, ∀t which implies that u∗(t) =
1, ∀t. This cannot be optimal given our assumption that b is
achievable with strict inequality in Proposition 4.1. Hence the
problem is normal and we set p∗0 = 1. If p∗ (0) = 0 then
p∗ (t) = 0, ∀t (see (15)). This would imply that for 0 ≤ t ≤ T ,
u∗ (t) = 0, which is clearly not optimal. Therefore, p∗ (0) > 0.
Let ψ(t) := p∗ (t)(1 − i∗ (t)) − 1. From (12) and (15) we get
ψ̇(t) = p∗ (t)(v − (i∗ (t) − 1)2 β)
(16)
Suppose βv > 1. Then for all 0 ≤ t ≤ T , ψ̇(t) > 0. If
p∗ (0) > 1 then this implies that u∗ (t) = 1, ∀t, which is not
optimal in view of our assumption regarding strict reachability
of b. Hence p∗ (0) < 1 and there exists 0 < t∗ < T such that
ψ(t) > 0 for t∗ < t ≤ T (otherwise u∗ (t) = 0, ∀t, which is
clearly not optimal). Therefore, u∗ (t) = 0 for 0 ≤ t ≤ t∗ and
u∗ (t) = 1 for t∗ < t ≤ T .
Now consider the case that βv < 1. Let ī := (1 − βv ). ī <
1. As in Section III, we denote by Ht∗ the Hamiltonian at
time t along the optimal state and control trajectory. Writing
(14) in a more suggestive form, we have Ht∗ = ψ(t)u∗ (t) −
βp∗ (t)i∗ (t)(i∗ (t) − ī). Recall that i∗ (0) = 0, i∗ (T ) = b and
note that i∗ (t) ↑ b. Therefore, there must exist 0 < t̂ < T such
that p∗ (t̂) > 0 and 0 < i∗ (t̂) < ī. Then Ht̂∗ > 0. Since the
Hamiltonian is constant for autonomous systems, Ht∗ > 0, ∀t.
Therefore, p∗ (0) > 1 (otherwise H0∗ = 0) and u∗ (0) = 1.
time such that 0 < t < t . Then u (t) = 0 and the value
function V (i, t) = t∗ − T . i(t∗ ) can be obtained in terms of
(i, t) as i(t∗ ) = 1−(1− ī )eī−βī(t∗ −t) . From (17) we obtain (via
1.0
0.8
0.6
0.4
0.0
0.2
Control value
0.8
0.6
0.4
Fraction of population
0.2
0.0
60
80
100
1.0
1.0
Fig. 1. Variation of the infected (i(t)) and the susceptible (s(t)) population
with time. The optimal strategy is to recruit initially till the budget is spent
and then stop. B = 2, v = 0.1, β = 0.05. By the end of time T = 10
(discretized into 100 steps), about 9% are still susceptible.
0.6
0.4
0.6
0.2
Control value
0.8
0.8
Susceptible
Infected
Control
0.0
ī
and, as earlier, y1 and y2
1−(1− iī )e−βīt∗
0
i − β1 = 0. Let i be a state on
denote the roots of i2 + 1+v−β
β
the optimal trajectory from i0 and let t be the corresponding
∗
∗
where, i(t∗ ) =
40
0.4
(17)
20
Time
Fraction of population
t∗ is given by the following nonlinear equation.
y1 − y2
= b,
y2 +
i(t∗)−y1 −β(y1 −y2 )(T −t∗ )
1 − i(t∗)−y2 e
0
0.2
means that ψ̇(t) < 0, ∀t (see (16)).
Now in this problem, a relatively simple form of (12) allows
us to derive value functions explicitly and, thereby, verify the
Hamilton-Jacobi-Bellman (HJB) equations. Since the algebra
is cumbersome, here we discuss only the case when βv > 1.
Suppose that the initial state is i0 and consider the control
(
0 0 ≤ t ≤ t∗
∗
u (t) =
1 t∗ < t ≤ T
Susceptible
Infected
Control
0.0
Then Hτ∗ > 0 implies that i∗ (τ ) < ī. Since i∗ (·) is monotonic,
the above implies that there can be at most two “crossings” of
ψ(·). Let t∗ and t̃ denote resp. the first time and the second
time ψ(·) = 0. Since ψ(0) > 0, continuity of ψ(·) implies that
u∗ (t) = 1 for 0 ≤ t ≤ t∗ . For t∗ < t ≤ t̃, ψ(t) < 0 hence
u∗ (t) = 0. For t̃ < t ≤ T, ψ(t) > 0 implying that u∗ (t) = 1.
∗
Suppose b ≥ ī. Then HT∗ > 0 implies that
q u (T ) = 1. Since
i∗ (t) ↑ b and ψ̇(t) > 0 for i∗ (t) > (1 − βv ), t̃ < T . When
q
b ≤ (1 − βv ), ψ(t) ≤ 0 for t∗ < t ≤ T since the condition
1.0
q
Observe from (16) that ψ̇(t) < 0 if i∗ (t) < (1 − βv ) and
q
ψ̇(t) > 0 if i∗ (t) > (1 − βv ). Let τ be such that ψ(τ ) = 0.
0
20
40
60
80
100
Time
i
(i,t)
= −g(i, t, t∗ ) iī2 and
the Implicit function theorem) ∂V∂i
∂V (i,t)
ī
= −g(i, t, t∗ )β ī(1 − i ) for some function g(i, t, t∗). It
∂t
is easy to see that
∂V (i, t)
∂V (i, t)
+ (−βi2 + (β − v)i)
=0
∂t
∂i
Thus, the HJB equation is indeed satisfied. Now if t∗ < t ≤ T ,
then u∗ (t) = 1 and the HJB equation for V (i, t) = t − T is
trivially verified.
V. N UMERICAL R ESULTS
In this section, we present numerical studies which validate
our theoretical results and provide some insights into them.
The numerical results are obtained by discretizing problems
(4) and (13), posing them as nonlinear constrained optimization problems and employing a gradient descent algorithm to arrive at an optimal solution. Time horizon T and
discretization step-size are fixed arbitrarily to 10 and 0.1
respectively. While discretization errors are apparent (and are
to be expected), our results seem to fit the theoretical analyses
quite well.
In Section III, we showed that the optimal control for
message dissemination is bang-bang, and the optimal strategy
Fig. 2. Variation of the infected (i(t)) and the susceptible (s(t)) population
with time. The optimal strategy is to recruit initially till the budget is spent
and then stop. B = 2, v = 0.1, β = 0.3. By the end of time T = 10
(discretized into 100 steps) only 2% are susceptible.
is to recruit “at full throttle” till the budget is spent and then
stop. The idea is get the population infected as fast as possible.
This is affirmed in Figures 1 and 2. We fix B = 2, v = 0.1
and consider two values of β- 0.05 and 0.3. It is well known
that an SIR epidemic occurs if βv < 1 [4]. Thus, even if the
budget is the same, when β = 0.3, an epidemic occurs in the
population and the fraction of population that has not received
the message till time T , i.e., s(T ), is only 2% (Figure 2)
whereas when β = 0.05, there is no “epidemic gain” and
s(T ) is 9% (Figure 1).
For diffusion of a topic, the optimal control is also bangbang. However, switching exhibited by it seems to depend on
v
β and the target buzz b. For an SIS epidemic system, it is
known that the disease persists if βv < 1 and dies down if
v
> 1. Mathematically, this means that in the former case
β
(s, i) = (1, 0) is globally asymptotically stable equilibrium
whereas in the latter case it is (s, i) = ( βv , 1 − βv ). Thus, when
1.0
0.5
1.0
0.8
0.8
0.4
0.6
Control value
0.4
0.3
0.2
Fraction of population
0.6
Control value
0.4
0.4
0.2
0.1
0.2
0.2
Fraction of population
0.6
0.8
Infected (buzz level)
Control
0
20
40
60
80
100
0.0
0.0
0.0
0.0
Infected (buzz level)
Control
0
20
40
60
80
100
Time
Fig. 3. Variation of the infected population with time (i(t)). v = 0.1, β =
0.05. The optimal strategy is to start recruiting just before the desired time
horizon.At T = 10 (discretized into 100 steps), the target buzz level of 80%
is reached.
Fig. 4. Variation of the infected populationp
with time (i(t)). v = 0.1, β =
v
)). The optimal strategy is
0.3. The target buzz level is 50%(> (1 −
β
to recruit for some initial duration and then stop. Time horizon is T = 10
(discretized into 100 steps).
1.0
0.8
Time
v
β
0.6
Control value
0.4
0.4
0.0
0.2
0.2
Fraction of population
0.6
0.8
Infected (buzz level)
Control
0.0
> 1, to create buzz b by T , it is optimal to start just ahead
of time otherwise it will fizzle out. This is demonstrated in
Figure 3. We set v = 0.1, β = 0.05 and b = 0.8.
On the other hand, if βv < 1, infecting the population
initially for some duration (t∗ ) ensures that the epidemic is
underway, and it is possible to reach the target buzz b “on its
way” to (1 − βv ) (this behavior of the optimal control may be
construed as exhibiting
a tipping point). This is indeed the case
q
when b ≤ (1 − βv ). If b > (1 − βv ), then an extra “push”
is required towards the end otherwise the buzz will end up
near the stable point of (1 − βv ). For numerical studies we set
v = 0.1, β = 0.3 and consider two targets- 0.5 and 0.8. These
parameters ensure that the targets are achievable in T = 10
(maximum reachable target is 0.927).
Shown in Figure 4 is the
q
case for b = 0.5. Note that (1− βv ) < b < (1− βv ). However,
only initial recruitment suffices in this case. In Figure 5,
b = 0.8 is considered (b > (1 − βv )) and the last push is
apparent.
0
20
40
60
80
100
Time
Fig. 5. Variation of the infected population with time (i(t)). v = 0.1, β =
0.3. The target buzz level is 80%(> (1 − βv )). The optimal strategy is to
recruit for some initial duration, stop and then recruit again till the end. Time
horizon is T = 10 (discretized into 100 steps).
R EFERENCES
VI. C ONCLUSION
In this paper we addressed two problems concerning information propagation in a population. We formulated them as
continuous-time deterministic optimal control problems and
derived the optimal controls. It turns out that in the case of
message dissemination, the optimal strategy is to recruit for
some initial duration till the budget is spent and then stop. For
creating buzz (diffusion of topic), the optimal strategy shows
different behaviors depending on whether the population can
sustain interest in the topic. The numerical results seem to
validate our theoretical analyses. This work can be extended
in two directions: explicitly modeling the network structure
to account for heterogeneities among agents and controlling
other parameters such as “retention rate” (v) in addition to
recruitment.
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